Ground states in non-relativistic quantum electrodynamics

Abstract

The excited states of a charged particle interacting with the quantized electromagnetic field and an external potential all decay, but such a particle should have a true ground state -one that minimizes the energy and satisfies the Schrödinger equation. We prove quite generally that this state exists for all values of the fine-structure constant and ultraviolet cutoff. We also show the same thing for a many-particle system under physically natural conditions. c 2000 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes. MGELML/3/2/01 2 exists (as far as we believe now) is the free particle (i.e., particle plus field). In the presence of an external potential, however, like the Coulomb potential of a nucleus, a ground state should exist.The difficulty in establishing this ground state comes from the fact that the bottom of the spectrum lies in the continuum (i.e., essential spectrum), not below it, as is the case for the usual Schrödinger equation. We denote the bottom of spectrum of the free-particle Hamiltonian for N particles with appropriate statistics by E 0 (N ). The 'free-particle' Hamiltonian includes the interparticle interaction (e.g., the Coulomb repulsion of electrons) but it does not include the interaction with a fixed external potential, e.g., the interaction with nuclei. When the latter is included we denote the bottom of the spectrum by E V (N ). It is not hard to see in many cases thatis, nevertheless, the bottom of the essential spectrum. The reason is that we can always add arbitrarily many, arbitrarily 'soft' photons that add arbitrarily little energy. It is the soft photon problem that is our primary concern here. The main point of this paper is to show how to overcome this infrared problem and to show, quite generally for a one-particle system, that a ground state exists for all values of the particle mass, the coupling to the field (fine-structure constant α = e 2 / c), the magnetic g-factor, and the ultraviolet cutoff Λ of the electromagnetic field frequencies, provided a bound state exists when the field is turned off. This result implies, in particular, that for a fixed ultraviolet cutoff renormalization of the various physical quantities will not affect the existence of a ground state. Of course, nothing can be said about the limit as the cutoff tends to infinity. We also include a large class of interactions much more general than the usual Coulomb interaction.The model we discuss has been used quite frequently in field theory. In its classical version it was investigated by Kramers [12] who seems to have been the first to point out the possibility of renormalization. The quantized version was investigated by Pauli and Fierz [24] in connection with scattering theory. Most importantly, it was used by Bethe [9] to obtain a suprisingly good value for the Lamb shift.Various restricted versions of the problem have been attacked successfully. In the early seventies Fröhlich investigated the infrared problem in translation invariant models of scalar elect...

Quasi-static Limits in Nonrelativistic Quantum Electrodynamics

“…Not pretending to be exhaustive, we refer the reader interested to these aspects to [3,8,11,12] and references therein.…”

Enhanced binding of an N-particle system interacting with a scalar bose field I

“…See e.g., [39] for the detail. The existence of a ground state of H PF has been established in [7,20] and its multiplicity in [28,33]. See also Hiroshima [29].…”

“…See also Hiroshima [29]. In particular Griesemer et al [20] proves the existence of a ground state for arbitrary values of coupling constant e, and they introduce a certain criterion for the existence of a ground state, which is called the binding condition. Our approach in this paper is indebted to the idea in [20], namely that our proof concentrates on checking the binding condition.…”

“…By extending the IMS localization argument to a quantum field model, the lowest two cluster threshold can be estimated. The following lemma is a simplified version of [20], since no interaction with a quantized radiation field exists.…”

“…Then it is hard to show the existence of ground states of K (α) with α = 0. Nevertheless the existence of ground states has been proven for various kinds of models with non-zero α in e.g., [2,7,20], where one fundamental assumption is that K 0 has a ground state.…”